Pre-Algebra Examples

x^{2} + 4 x + 29 = 0

$x^{2} + 4 x + 29 = 0$

Step 1

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values

a = 1

$a = 1$ ,

b = 4

$b = 4$ , and

c = 29

$c = 29$ into the quadratic formula and solve for

x

$x$ .

\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

4

$4$ to the power of

2

$2$ .

x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 3.1.2

Multiply

- 4 \cdot 1 \cdot 29

$- 4 \cdot 1 \cdot 29$ .

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Step 3.1.2.2

Multiply

- 4

$- 4$ by

29

$29$ .

x = \frac{- 4 \pm \sqrt{16 - 116}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

x = \frac{- 4 \pm \sqrt{16 - 116}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Step 3.1.3

Subtract

116

$116$ from

16

$16$ .

x = \frac{- 4 \pm \sqrt{- 100}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 3.1.4

Rewrite

- 100

$- 100$ as

- 1 (100)

$- 1 (100)$ .

x = \frac{- 4 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 3.1.5

Rewrite

\sqrt{- 1 (100)}

$\sqrt{- 1 (100)}$ as

\sqrt{- 1} \cdot \sqrt{100}

$\sqrt{- 1} \cdot \sqrt{100}$ .

x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

x = \frac{- 4 \pm i \cdot \sqrt{100}}{2 \cdot 1}

$x = \frac{- 4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 3.1.7

Rewrite

100

$100$ as

10^{2}

$10^{2}$ .

x = \frac{- 4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}

$x = \frac{- 4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

x = \frac{- 4 \pm i \cdot 10}{2 \cdot 1}

$x = \frac{- 4 \pm i \cdot 10}{2 \cdot 1}$

Step 3.1.9

Move

10

$10$ to the left of

i

$i$ .

x = \frac{- 4 \pm 10 i}{2 \cdot 1}

$x = \frac{- 4 \pm 10 i}{2 \cdot 1}$

Step 3.2

Multiply

$2$ by

$1$ .

$x = \frac{- 4 \pm 10 i}{2}$

Step 3.3

Simplify

$\frac{- 4 \pm 10 i}{2}$ .

$x = - 2 \pm 5 i$

Step 4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

$4$ to the power of

$2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 4.1.2

Multiply

$- 4 \cdot 1 \cdot 29$ .

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Step 4.1.2.1

Multiply

$- 4$ by

$1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Step 4.1.2.2

Multiply

$- 4$ by

$29$ .

$x = \frac{- 4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Step 4.1.3

Subtract

$116$ from

$16$ .

$x = \frac{- 4 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 4.1.4

Rewrite

$- 100$ as

$- 1 (100)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 4.1.5

Rewrite

$\sqrt{- 1 (100)}$ as

$\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 4.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 4.1.7

Rewrite

$100$ as

$10^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 4 \pm i \cdot 10}{2 \cdot 1}$

Step 4.1.9

Move

$10$ to the left of

$i$ .

$x = \frac{- 4 \pm 10 i}{2 \cdot 1}$

Step 4.2

Multiply

$2$ by

$1$ .

$x = \frac{- 4 \pm 10 i}{2}$

Step 4.3

Simplify

$\frac{- 4 \pm 10 i}{2}$ .

$x = - 2 \pm 5 i$

Step 4.4

Change the

$\pm$ to

$+$ .

$x = - 2 + 5 i$

Step 5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise

$4$ to the power of

$2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 5.1.2

Multiply

$- 4 \cdot 1 \cdot 29$ .

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Step 5.1.2.1

Multiply

$- 4$ by

$1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Step 5.1.2.2

Multiply

$- 4$ by

$29$ .

$x = \frac{- 4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Step 5.1.3

Subtract

$116$ from

$16$ .

$x = \frac{- 4 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 5.1.4

Rewrite

$- 100$ as

$- 1 (100)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 5.1.5

Rewrite

$\sqrt{- 1 (100)}$ as

$\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 5.1.7

Rewrite

$100$ as

$10^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 4 \pm i \cdot 10}{2 \cdot 1}$

Step 5.1.9

Move

$10$ to the left of

$i$ .

$x = \frac{- 4 \pm 10 i}{2 \cdot 1}$

Step 5.2

Multiply

$2$ by

$1$ .

$x = \frac{- 4 \pm 10 i}{2}$

Step 5.3

Simplify

$\frac{- 4 \pm 10 i}{2}$ .

$x = - 2 \pm 5 i$

Step 5.4

Change the

$\pm$ to

$-$ .

$x = - 2 - 5 i$

Step 6

The final answer is the combination of both solutions.

$x = - 2 + 5 i, - 2 - 5 i$